Preamble

The functions of space and time (i.e., fields) used to model material behaviour take values which may be real numbers, vectors, or higher-order tensors. Formal manipulations of tensors (i.e., tensor algebra) are best understood in terms of vector spaces. Here basic concepts and results are reviewed for completeness and for establishing familiarity with the notation employed. Vectorial entities (i.e., entities which have both direction and magnitude and combine like displacements) are modelled in terms of a three-dimensional inner-product vector space V, and higher-order tensorial entities are described in terms of algebraic constructs of V.

Simple considerations of rectilinear changes of position (i.e., displacements) and the notion of perpendicularity are used to establish the three-dimensional inner product vector space V used to model vectorial quantities, irrespective of their physical dimensions of mass, length, and time, and units of measurement. Linear transformations on V are defined and shown to have algebraic features in common with V, so motivating the definition of a general abstract vector space. The transpose of a linear transformation L on V and the tensor product of two vectors are defined without recourse to basis-dependent representations: such representations are derived upon selecting an orthonormal basis for V. Criteria which establish the invertibility or otherwise of a linear transformation L on V are identified, and the principal invariants and characteristic equation of L are analysed using alternating trilinear forms on V.

Motivation

Material behaviour at length scales greatly in excess of molecular dimensions (i.e., macroscopic behaviour) is usually modelled in terms of the continuum viewpoint. From such a perspective the matter associated with any physical system (or body) of interest is, at any instant, considered to be distributed continuously throughout some spatial region (deemed to be the region ‘occupied’ by the system at this instant). Reproducible macroscopic phenomena are modelled in terms of deterministic continuum theories. Such theories have been highly successful, particularly in engineering contexts, and include those of elasticity, fluid dynamics, and plasticity. The totality of such theories constitutes (deterministic) continuum mechanics. The link between actual material behaviour and relevant theory is provided by experimentation/observation. Specifically, it is necessary to relate local experimental measurements to continuum field values. However, the value of any local measurement made upon a physical system is the consequence of a local (both in space and time) interaction with this system. Further, local measurement values exhibit erratic features if the scale (in space-time) is sufficiently fine, and such features become increasingly evident with diminishing scale. Said differently, sufficiently sensitive instruments always yield measurement values which fluctuate chaotically in both space and time (i.e., these values change perceptibly, in random fashion, with both location and time), and the ‘strength’ of these fluctuations increases with instrument sensitivity (i.e., with increasingly fine-scale interaction between instrument and system).

Preamble

The continuum viewpoint is consistent with our physical prejudices, engendered by sensory evidence. However such a perspective gives rise to some fundamental conceptual and physical difficulties which involve questions of scale, interpretation, and reproducibility of phenomena. Here we outline several of these difficulties and indicate how one is forced to take account of the fundamentally discrete nature of matter and the spatial scales at which physical systems are monitored.

The Natural Continuum Prejudice

We unconsciously adopt a continuum viewpoint when observing and interacting with the world about us. For example, we regard the air we breathe as tangible (when feeling the wind on our faces or filling our lungs) and as permeating the space about us. If we pour some water into a glass, then this water appears to take up a definite shape, determined by the sides and base of the glass and by the free water surface. The water seems to fill this shape, apart from possible visible bubbles of trapped air or immersed foreign particles. The glass itself appears to occupy a definite region, delineated by its bounding surfaces, with the possible exception of visible imperfections. However, while we can see ‘inside’ water and glass, this is not the case for opaque objects, for which only the external boundary is amenable to direct observation. Nevertheless, we often regard opaque objects to be full of matter in the sense of occupying all space within their perceived external boundaries.

Preamble

Continuum mechanics is the study of material behaviour as manifest at macroscopic scales of length and time. Irrespective of molecular constitution, or whether the material is gaseous, liquid, or solid, mathematical descriptions of such behaviour have a common foundation. This foundation has been developed in Chapters 3 through 8, and is codified in terms of the system of balance relations for mass, linear and rotational momentum, and energy, and the physical descriptors (i.e., fields) which appear therein. Such relations, which serve as evolution equations for mass etc., involve terms directly related to molecular behaviour, but take the same form regardless of the specific and explicit scales of length and time associated with the averaging procedure.

In order to model a particular material system, with the aim of predicting its behaviour under prescribed circumstances, it is necessary

1. (i) to distinguish this behaviour from that of other materials subjected to the same prescribed circumstances, and

2. (ii) to make precise just what is intended by ‘prescribed circumstances’.

In respect of prescription (i), the nature (or ‘constitution’) of an individual system is identified in terms of constitutive relations which specify the ‘response’ of the material to the consequences of prescribed circumstances. Prescription (ii) involves specification of agencies external to the system (including the effect of gravity and of contact between the system and the exterior world across its boundary) together with initial information.

Preamble

Time averaging is shown here to pervade the microscopic foundations of macroscopic modelling, whether in appreciating the nature of intermolecular forces in terms of subatomic dynamics (recall Section 5.4), establishing links between continuum field values and actual measurements (recall Section 3.10), deriving balance relations for systems with changing material content, or exploring the conceptual foundations of statistical mechanics.

After recalling the notion of a Δ-time average introduced in Section 5.4, such averaging is applied directly to the continuity equation and balances of linearmomentum and energy established in Chapters 4, 5, and 6. Field values in the resulting relations are thereby identified as local averages of molecular quantities computed jointly in both space and time. Such relations apply to material systems which do not change with time. Any system M+ whose material content does change with time is regarded to be a subset of an unchanging material system M. Relations for M+ are established by introducing a membership function ei for each element Pi of M: ei(t) = 1 (or 0) according to whether Pi ∊ M+ (or Pi ∉ M+) at time t. An equation which prescribes the time evolution of a single global representative velocity for M+ is derived utilising time averaging: this serves to establish the methodology to be used later in deriving local balances of mass, linear momentum, and energy, and also to discuss the fundamentals of rocket and jet propulsion.

Preamble

Upon modelling molecules as point masses, volumetric densities ρw of mass and pw of momentum are defined as local spatial averages of molecular masses and momenta using a weighting function w which, while possessing certain essential features, is otherwise unspecified and general. Partial (time) differentiation of ρw yields the continuity equation (2.5.16) in which the velocity field vw ≔ pw/ρw. The physical interpretations of ρw,pw and vw depend crucially upon the choice of w. Several physically distinguished classes of weighting function are discussed. Emphasis is placed upon a particular class because the corresponding interpretations of the mass density and velocity fields, and of the boundary, associated with any body are particularly simple. The conceptual problems C.P.1, C.P.2, and C.P.3 listed in Section 3.8 are addressed and completely resolved.

Weighted Averages and the Continuity Equation

The mass density ρ(x, t) at a given location x (a geometrical point) and time t is a local measure of ‘mass per unit volume’. The key questions here are ‘What mass?’ and ‘What volume?’

The mass of any given body of matter derives ultimately from that of its constituent fundamental discrete entities (i.e., electrons and atomic nuclei). While any such fundamental entity could be modelled as a point mass whose location is that of its mass centre, for the purposes of this chapter we adopt a molecular viewpoint. Specifically, we choose here to regard a material system (or body) ℳ to be a fixed, identifiable set of (N, say) molecules modelled as point masses.

Preamble

In this chapter we address fundamental aspects of continuum modelling in respect of kinematics, mass conservation, balances of linear and rotational momentum, and balance of energy.

After considering the role of mass density in modelling the presence of ‘matter’, we discuss the manner in which the detailed macroscopic distortion of any material body can be monitored. This is markedly different for solids and fluids, but in both cases it is possible to motivate the notion of material point and thereby establish basic kinematic concepts such as deformation, motion, and velocity. The formal (axiomatic) approach to kinematics is outlined for comparison. Mass conservation is motivated for solids and postulated to hold in general. Dynamical considerations are first addressed for a body as a whole. In addition to tractions on boundaries, the possibility of surface and body couples is considered. Global balances of linear and rotational momentum are postulated and applied to rigid bodies both to emphasise their often-neglected status as a special case of material continua and to develop familiarity with notation, concepts and basic manipulations. Local forms of balance are derived in standard fashion by postulating balances for matter in arbitrary subregions of the region instantaneously occupied by the body, invoking a transport theorem, and then establishing the existence of stress and couple stress tensors and a heat flux vector. It is these local forms of balance that can be derived directly from molecular considerations using the weighting function methodology to be introduced in Chapter 4.

Preamble

Analternative strategy for relating continuum field values to their microscopic origins is outlined. Spatial averaging of additive molecular quantities is effected in terms of ‘cells’. In contrast to weighting function methodology, linear momentum balance for macroscopic regions is established before deducing the local form. The existence of a traction field on the boundary of such regions is derived via the assumption of short-range molecular interactions. The corresponding interaction stress tensor is obtained in the standard manner of continuum mechanics. Unlike balances obtained in terms of weighting functions, which hold for any given pair (∈, Δ) of length–time scales, fields obtained as cellular averages exist only if their values are somewhat insensitive to changes in ∈, Δ, and cell shape.

Cellular Averaging

Recall Subsection 4.4.1 in which selection of an appropriate weighting function [namely a mollified version of relations (4.4.4)] delivered spatial cellular averages. The analyses of Sections 4.2, 5.2 through 5.6, 6.2 and 6.3, 7.2 through 7.5, 8.4 through 8.6, 8.9 and 9.2 through 9.7 apply to such a choice. Thus, in particular, the standard relations [which express mass conservation (4.2.16), momentum balance (2.7.30) with T = Tw, where Tw is given by (5.5.20), and energy balance (6.2.75)] hold with field values defined in terms of cellular molecular averages. The associated (i.e., cell-based) notion of material point, and motion thereof, derives from the corresponding velocity field vw defined in Section 5.2.

This volume is the result of a workshop, “Partial Differential Equations and Fluid Mechanics”, which took place in the Mathematics Institute at the University of Warwick, June 15th–19th, 2010.

Several of the speakers agreed to write review papers related to their contributions to the workshop, while others have written more traditional research papers. We believe that this volume therefore provides an accessible summary of a wide range of active research topics, along with some exciting new results, and we hope that it will prove a useful resource for both graduate students new to the area and to more established researchers.

We would like to express their gratitude to the following sponsors of the workshop: the London Mathematical Society, EPSRC (via a conference grant EP/I001050/1), and the Warwick Mathematics Department. JCR is currently supported by an EPSRC Leadership Fellowship (grant EP/G007470/1).

Finally it is a pleasure to thank Yvonne Collins and Hazel Higgens from the Warwick Mathematics Research Centre for their assistance during the organization of the workshop.